The current paper is devoted to the study of pullback attractors for
general nonautonomous and random parabolic equations on non-smooth
domains $D$. Mild solutions are considered for such equations. We
first extend various fundamental properties for solutions of smooth
parabolic equations on smooth domains to solutions of general
parabolic equations on non-smooth domains, including continuous
dependence on parameters, monotonicity, and compactness, which are
of great importance in their own. Under certain dissipative
conditions on the nonlinear terms, we prove that mild solutions
with initial conditions in $L_q(D)$ exist globally for $q$ » $1$. We
then show that pullback attractors for nonautonomous and random
parabolic equations on non-smooth domains exist in $L_q(D)$ for
$1$ « $q$ < $\infty$.